a first Approximation to the Orbit of a Comet. 183 
r* = R"- -1- 2R cos. A cos. (^e — c) f 
^^=1 R'" -f. 2R' cos. x' cos. — c') . 
V = RR' cos. — + R' cos. A cos. ^ 
-}" cos* cos. ('6’ — c') . ^' + cos. 7 . 
3dly. We must employ the same final equation as in the 
last method, to determine upon which the values of /'% 
and V all depend. 
Application to the first Comet of 1805. 
If we turn to the calculation of the orbit of this comet, by 
the former method it will appear that the intersection of the 
great circle which passes through the two extreme places of 
the comet with the ecliptic is very near the place of the earth 
at the middle observation ; for the distance of these two points, 
or the arc e ° — ?7, amounts to no more than 2° 21' i 6 "~. This 
instance therefore is very near the limit wlien the former 
solution becomes indeterminate ; which takes place when the 
arc — n is evanescent, or when the points abovementioned 
coincide. In these circumstances we can hardly expect that 
the orbit will be determined with much certainty : for, on ac- 
count of the small divisor tan. the values of and 
the angle w, will sulfer considerable variations when small 
changes are made in the observed places of the comet, more 
particularly in the mean place. This comet therefore furnishes 
a very proper example for applying the method we have just 
investigated. 
Using the same observations as in the former calculation, 
we shall get, 
log, 46*= 9-2341873 
